Properties

Factorization system

Every field extension can be factorized as a purely transcendental extension? followed by an algebraic extension. Indeed, by Zorn's lemma, we may construct a transcendence basis (i.e. maximal algebraically independent set) BB, and the purely transcendental part is the subfield generated by BB.

Unfortunately, this does not yield an orthogonal factorization system: given a field KK, we may form the field K(x)K (x) of rational functions over KK, which is a purely transcendental extension of KK, and we may form the algebraic closureK(x)¯\overline{K (x)}, which is an algebraic extension of K(x)K (x); but we have the following commutative diagram,

where K(x2)K (x^2) is the subfield of K(x)K (x) generated by x2x^2, and K(x2)→K(x)¯K (x^2) \to \overline{K (x)} is algebraic, yet there is no homomorphism K(x)→K(x2)K (x) \to K (x^2) making both evident triangles commute.